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Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is increasing if and only if:

$\forall x, y \in S: x \preceq_1 y \ \implies \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

Note that this definition also holds if $S = T$.

Also known as

An increasing mapping is also known as order-preserving, isotone and non-decreasing.

Some authors refer to this concept as a monotone mapping, but that term has a different meaning on ProofWiki.

Also defined as

Some sources insist at the point of definition that $\phi$ be an injection for it to be definable as order-preserving, but this is conceptually unnecessary.

Also see